Coordinate-space representation

In Eq. (109), y denotes an n-dimensional diagonal matrix, with element y being the width parameter for the coherent state of the y th dimension. The coordinate space representation of an n-dimensional coherent state is the product of n onedimensional minimum uncertainty wave packets... 

The 1- and 2-RDMs can be obtained in the coordinate-space representation via the expansion theorem ... 

Carrying out the Fourier transformation, one obtains the coordinate space representation for the third Zemach moment, in terms of the weighted convolution of two nuclear charge densities p r) [18]... 

Similarly to (7.1) it is easy to write a coordinate space representation for the perturbation potential corresponding to the diagram in Fig. 7.10... 

The key feature to making a semiclassical approach practical is to avoid having to deal explicitly with the double-ended boundary conditions in Eq. (3.3) [16-20]. (The initial condition x(X, p( 0) = X is obviously easy to deal with.) To do this, one uses the standard coordinate space representation of Eq. (2.5),... 

Given this correspondence between the matrix elements of a partially Wigner transformed operator in the subsystem and mapping bases, we can express the quantum-classical Liouville equation in the continuous mapping coordinates [53]. The first step in this calculation is to introduce an n-dimensional coordinate space representation of the mapping basis,... 

In Section 4.2.2, we used the displacement (translation) operator exp(—ibpi/h). We consider here this operator and its action on the state ), i.e., we consider the momentum-space and coordinate-space representations of b) = exp(—ibpi/h) ). 

I Here there are two different possibilities. If the physics is such that E, n 1) is a solution of a single-channel problem, then its coordinate space representation can a(ways be written [8] as... 

This ensures that we can replace the p and p with the coordinate space representation —iV of the momentum operator p, since... 

Here, the Hilbert space vector xj/j) has a coordinate space representation xl/j(q)= q l/j), and xl/j(q) is the probability of observing a given value of q when the system is in a state defined by energy Ej. In general, the system may be degenerate, in which case several Ej may have the same numerical value. 

In the above discussion of relaxation to equilibrium, the density matrix was implicitly cast in the energy representation. However, the density operator can be cast in a variety of representations other than the energy representation. Two of the most connnonly used are the coordinate representation and the Wigner phase space representation. In addition, there is the diagonal representation of the density operator in this representation, the most general fomi of p takes the fomi... 

We can now proceed to the generation of conformations. First, random values are assigne to all the interatomic distances between the upper and lower bounds to give a trial distam matrix. This distance matrix is now subjected to a process called embedding, in which tl distance space representation of the conformation is converted to a set of atomic Cartesic coordinates by performing a series of matrix operations. We calculate the metric matrix, each of whose elements (i, j) is equal to the scalar product of the vectors from the orig to atoms i and j ... 

At the end of Section 8.16 we mentioned that the Fock representation avoids the use of multiple integrations of coordinate space when dealing with the many-body problem. We can see here, however, that the new method runs into complications of its own To handle the immense bookkeeping problems involved in the multiple -integrals and the ordered products of creation and annihilation operators, special diagram techniques have been developed. These are discussed in Chapter 11, Quantum Electrodynamics. The reader who wishes to study further the many applications of these techniques to problems of quantum statistics will find an ample list of references in a review article by D. ter Haar, Reports on Progress in Physics, 24,1961, Inst, of Phys. and Phys. Soc. (London). 

We have carried out tins discussion in occupation number representation or coordinate representation each with a definite number N of particles. Similar results follow for the Fock space representation and the properties of grand ensembles. Averages over grand ensembles are also independent of time when the probabilities > are independent of time, whether the observable commutes with H or not. 

Every n vector can be represented as a point in an -dimensional coordinate space. The n elements of the vector are the coordinates along n basis vectors, such as defined in the previous section. The null vector 0 defines the origin of the coordinate space. Note that the origin together with an endpoint define a directed line segment or axis, which also represents a vector. Hence, there is an equivalence between points and axes, which can both be thought as geometrical representations of vectors in coordinate space. (The concepts discussed here are extensions of those covered previously in Sections 9.2.4 to 9.2.5.)... 

In coordinate space, the diagonal elements of the canonical density matrix in the Fourier path integral representation are given by [20]... 

The representation of this expression in the coordinate space can be done as usual. 

This section provides a brief discussion on the construction of reduced-dimension chemistry spaces for sets of molecules described by coordinate-free or by high-dimensional coordinate-based representations. Inherently low-dimensional... 

Suppose that (xi,X2,..., xn) is the position-space representation of the N-electron wavefunction. It is a function of the space-spin coordinates Xk = (Ofcj ctjfc) in which is the position vector of the kth electron and Gk is its spin coordinate. The position-space wavefunction is obtained by solving the usual position- or r-space Schrodinger equation by one of the many well-developed approximate methods [32-34]. 

Fig. 3. Mechanistic pathways are depicted connecting the initial and final states of two categories of quantum systems. In case (a), file system has a modest number of levels and hence, a discrete representation is appropriate. The three of many possible pathways shown connecting the initial and final states would each have an associated amplitude. In case (b), a bond or coordinate space picture is a more natural perspective to consider for understanding the pathway to products, in this case, dissociation. The various light-shaded bonds indicate examples of possible locations where tailored radiation might enter to control the dynamics.

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